Lecture 7 Circuits Ch. 27 Cartoon -Kirchhoff's Laws Topics –Direct Current Circuits –Kirchhoff's Two Rules –Analysis of Circuits Examples –Ammeter and voltmeter –RC circuits Demos –Three bulbs in a circuit –Power loss in transmission lines –Resistivity of a pencil –Blowing a fuse Elmo
Kirchhoff's Laws 1.The sum of the potential drops around a closed loop is zero. This follows from energy conservation and the fact that the electric field is a conservative force. 2. The sum of currents into any junction of a closed circuit must equal the sum of currents out of the junction. This follows from charge conservation.
Example (Single Loop Circuit) How do we apply Kirchhoff’s Rule 1? Must assume the direction of the current – assume clockwise. Choose a starting point and sum all voltages drops around the circuit using Ohm’s Law
Example (Single Loop Circuit) a.Potential drop across resistors is negative when traversing in the same direction as current. b.Sign of E for a battery is positive when traversing in the same direction as current and assumed current direction is negative to positive in the battery. c. Sign of E for a battery is negative when traversing in the same direction as current and assumed current direction is positive to negative in the battery. d. Reverse the sign for b and c when you are traversing the circuit in the opposite direction as the assumed current.
Now let us put in numbers. Suppose: amp Suppose: amp We get a minus sign. It means our assumed direction of current must be reversed. Note that we could have simply added all resistors and get the R eq. and added the EMFs to get the E eq. And simply divided. amp Sign of EMF Battery 1 current flows from - to + in battery +E 1 Battery 2 current flows from + to - in battery -E 2 In 1 the electrical potential energy increases In 2 the electrical potential energy decreases
Example with numbers Quick solution: Question: What is the current in the circuit? Write down Kirchhoff’s loop equation. Loop equation Assume current flow is clockwise. Do the batteries first – Then the current.
Example with numbers (continued) Question: What are the terminal voltages of each battery? 2V: 4V: 12V:
Multiloop Circuits Kirchoff’s Rules 1. in any loop 2. at any junction Find i, i 1, and i 2 Rule 1 – Apply to 2 loops (2 inner loops) 1. 2. Rule 2 3. We now have 3 equations with 3 unknowns. multiply by 2 multiply by 3 subtract them Find the Joule heating in each resistor P=i 2 R. Is the 5V battery being charged?
Method of determinants for solving simultaneous equations Cramer’s Rule says if : Then,
You try it for i 1 and i 2. See inside of front cover in your book on how to use Cramer’s Rule. For example solve for i Method of determinants using Cramers Rule and cofactors Also use this to remember how to evaluate cross products of two vectors.
Another example Find all the currents including directions. Loop 1 Loop 2 Loop 1 Loop 2 ii i i i1i1 i2i2 i2i2 Multiply red eqn of loop 1 by 2 and subtract from the red eqn of loop 2 Start here
Rules for solving multiloop circuits 1.Replace series resistors or batteries with their equivalent values. 2.Choose a direction for i in each loop and label diagram. 3.Write the junction rule equation for each junction. 4.Apply the loop rule n times for n interior loops. 5.Solve the equations for the unknowns. Use Cramer’s Rule if necessary. 6.Check your results by evaluating potential differences.
3 bulb question The circuit above shows three identical light bulbs attached to an ideal battery. If the bulb#2 burns out, which of the following will occur? a)Bulbs 1 and 3 are unaffected. The total light emitted by the circuit decreases. b)Bulbs 1 and 3 get brighter. The total light emitted by the circuit is unchanged. c)Bulbs 1 and 3 get dimmer. The total light emitted by the circuit decreases. d)Bulb 1 gets dimmer, but bulb 3 gets brighter. The total light emitted by the circuit is unchanged. e)Bulb 1 gets brighter, but bulb 3 gets dimmer. The total light emitted by the circuit is unchanged. f)Bulb 1 gets dimmer, but bulb 3 gets brighter. The total light emitted by the circuit decreases. g)Bulb 1 gets brighter, but bulb 3 gets dimmer. The total light emitted by the circuit decreases. h)Bulb 1 is unaffected, but bulb 3 gets brighter. The total light emitted by the circuit increases. i)None of the above.
When the bulb #2 is not burnt out: For Bulb #1 For Bulb #2 For Bulb #3
For Bulb #1 For Bulb #2 For Bulb #3 So, Bulb #1 gets dimmer and bulb #3 gets brighter. And the total power decreases. f) is the answer. Before total power was After total power is When the bulb #2 is burnt out:
How does a capacitor behave in a circuit with a resistor? Charge capacitor with 9V battery with switch open.. Remove battery and close the switch. What happens?
Discharging a capacitor through a resistor V(t) Potential across capacitor = V = just before you throw switch at time t = 0. Potential across Resistor = iR Just after you throw the switch What is the current I at time t?
Integrating both the sides Time constant =RC What is the charge Q at time t?
How the charge on a capacitor varies with time as it is being charged What about charging the capacitor? Same as before Note that the current is zero when either the capacitor is fully charged or uncharged. But the second you start to charge it or discharge it, the current is maximum.
Instruments Galvanometers:a coil in a magnetic field that senses current. Ammeters:measures current. Voltmeter:measures voltage. Ohmmeters:measures resistance. Multimeters:one device that does all the above. Galvanometer is a needle mounted to a coil that rotates in a magnetic field. The amount of rotation is proportional to the current that flows through the coil. Symbolically we write Usually when
Ohmmeter Adjust R s so when R=0 the galvanometer read full scale.
Ammeter The idea is to find the value of R S that will give a full scale reading in the galvanometer for 5A Ammeters have very low resistance when put in series in a circuit. You need a very stable shunt resistor. Very small
Voltmeter Use the same galvanometer to construct a voltmeter for which full scale reading in 10 Volts. What is the value of R S now? So, the shunt resistor needs to be about 20K Note: the voltmeter is in parallel with the battery. We need
Chapter 27 Problem 19 In Figure 27-34, R1 = 100, R2 = 30, and the ideal batteries have emfs script e1 = 6.0 V, script e2 = 5.0 V, script e3 = 3.0 V. Fig. 27-34 (a) Find the current in resistor 1. (b) Find the current in resistor 2. (c) Find the potential difference between points a and b. Fig. 27-34
Chapter 27 Problem 27 In Figure 27-40, the resistances are R1 = 0.5, R2 = 1.7, and the ideal batteries have emfs script e1 = 2.0 V, and script e2 = script e3 = 3 V. Fig. 27-40 (a) What is the current through each battery? (Take upward to be positive.) battery 1 battery 2 battery 3 (b) What is the potential difference Va - Vb?
Chapter 27 Problem 38 A simple ohmmeter is made by connecting a 4.0 V battery in series with a resistance R and an ammeter that reads from 0 to 1.00 mA, as shown in Figure 27-47. Resistance R is adjusted so that when the clip leads are shorted together, the meter deflects to its full-scale value of 1.00 mA. Fig. 27-47 (a) What external resistance across the leads results in a deflection of 10% of full scale? (b) What resistance results in a deflection of 50% of full scale? (c) What resistance results in a deflection of 90% of full scale? (d) If the ammeter has a resistance of 40.0 and the internal resistance of the battery is negligible, what is the value of R?